Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior

Abstract

This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g\not\equiv 0(\mathrm{mod}6)$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface ${\Sigma }_g$ of genus $g$ is achieved by the monodromy of some ${\Sigma }_g$-bundle over the circle obtained from $N(\frac{3}{-2})$ or $N(\frac{1}{-2})$ by Dehn filling two cusps. These minimizers are the same ones identified by Hironaka, Aaber-Dunfiled, Kin-Takasawa independently. In the case $g\equiv 6(\mathrm{mod}12)$ we find a new family of pseudo-Anosovs defined on ${\Sigma }_g$ with orientable invariant foliations obtained from N(-6) or N(4) by Dehn filling two cusps. We prove that if ${\delta }_g^{+}$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on ${\Sigma }_g$, then $$\underset{\begin{array}{c}g\equiv 6(\mathrm{mod}12)g\to \infty \end{array}}{\mathrm{limsup}}g\log {\delta }_g^{+}\le 2\log \delta (D_5)\approx 1.0870,$$ where $\delta (D_n)$ is the minimal dilatation of pseudo-Anosovs on an $n$-punctured disk. We also study monodromies of fibrations on N(1). We prove that if ${\delta }_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with $n$ punctures, then $$\underset{n\to \infty }{\mathrm{limsup}}n\log {\delta }_{1,n}\le 2\log \delta (D_4)\approx \mathrm{1.6628.}$$

46 pages, 14 figures; version 3: Major change in Section 2.1, and minor corrections

References in corpus (2)

• A family of pseudo-Anosov braids with small dilatation
• Small dilatation pseudo-Anosovs and 3-manifolds

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• An algorithm to compute the Teichmueller polynomial from matrices
• The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of the minimal pseudo-Anosovs dilatations